Apart from the immediate implications for the Casimir effect, astrophysics, and cosmology, the high-temperature expansion of the free energy for fermions can be used to analyze the derivative expansion of the one-loop contribution to the effective action at zero temperature (see, e.g., [1]}, [2]}) regularized by the energy cutoff. According to the general prescriptions of the renormalization theory [3]}, the structures appearing as the divergencies in the effective action must be included to the initial action to cancel these divergencies (not necessarily to zero). As a rule, these divergencies and the finite part depend nontrivially on the Killing vector \(\xi ^\mu \)  defining the stationarity of the background and the vacuum state of the quantum fields ([4]}, see also [5]}, [6]}, [7]}). We shall see in the present paper that this is indeed the case. Hence, the analysis of the high-temperature expansion sheds a light on the low energy quantum dynamics of the vector field \(\xi ^\mu \)  as it was discussed in [2]}, [4]}. To see how this works, one may bear in mind the expansion of the Heisenberg-Euler effective action [10]}, [11]}, where the first non-trivial term describes the light by light scattering.
